Update: March 26, the name of the post originally entitled “Polymath1: Probable Success!” was now updated to “Polymath1: Success!” It is now becoming clear that there are three (perhaps four) new emerging proofs of DHJ. (April 2: See this post by Terry Tao. As this update was also based on briefly talking with Terry, Terry’s new post gives a better description on the state of affairs and relations between the different proofs.)

The proof directly emerged from the project indeed looks conceptually different and simpler than all other proofs, and may indeed lead to the simplest known proof for Szemeredi’s theorem. (But for this we will have to wait for the details.) In addition, there is a new Ergodic proof by Tim Austin, which was partially inspired by and which used (among several other ingredients developed by Austin) some ideas and results discovered in the polymath project. Both the original ergodic proof and Austin’s proof were (at least roughly) “combinatorialized”.

In what sense was it a massive open collaboration? It is true that in the crucial phases of polymath, the phases where two concrete strategies for proofs were considered, the number of pivotal participants was not large. But there was an initial phase were probably more than a hundred mathematicians took part as observers and as commentators. The comments in this early phase played some role in the later developments but what is more important is that this stage have led to the (emerging) selection of the team that developed the proof. Among the hundreds, those who felt they have ideas that can be crucial, or methods that could be helpful, and were smart or lucky to be correct, and had the persistence to follow these ideas and how these ideas can be combined with other ideas, became the pivotal players. The team that played the game was not so large, but the main massive ingredient of the project which accounts for its accessive mathematical power was in the “draft”. The team emerged from a massive number of participants. (So if you believe there were 10 pivotal players out of a hundred, think about the emergence of the team among possible (but, of course, not equally plausible) teams, as a point were polymath had extra power.)

Two related posts: Tim Gowers raised inthis post interesting questions regarding the possibility of projects were the actual number of provers will be massive. Here on my blog we have a post with an “open discussion” on what are the correct bounds for Roth type problems. The emphasis is on “small-talk discussion” and not on actual “hard-nose researching”.

We took the opportunity to spend three days of “Purim” visiting northern Israel. Coming back I saw two new posts on Tim Gowers’s blog entitled “Problem solved probably” and “polymath1 and open collaborative mathematics.” It appears that “polymath1” has led to a new proof for the density version of Hales-Jewett’s theorem for k=3 which was the original central goal! Also it looks like the open collaboration mode (while not being a massive collaboration) was indeed useful.

Perhaps the most important thing is to make sure that a complete proof for the k=3 case is indeed in place (as these famous problems sometimes “fight back,” as Erdos used to say). The outline is described here. If everything is OK as Tim and other participants expect, there are already some discussions or even plans about an extension to the general k case. This seems to be the next major step in the project. There are also other fruits from the various threads of the polymath1 project. Overall, this looks very exciting! The mathematical result is a first-rate achievement, and the mode of cooperation is novel, interesting and appears genuinely useful.

Let me quote what Tim writes about it:“Better still, it looks very much as though the argument here will generalize straightforwardly to give the full density Hales-Jewett theorem. We are actively working on this and I expect it to be done within a week or so. (Work in progress can be found on the polymath1 wiki.) Better even than that, it seems that the resulting proof will be the simplest known proof of Szemerédi’s theorem.”sababa!